find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{25x^{2}-64}{15x + 24}$
find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{25x^{2}-64}{15x + 24}$
Answer
Explanation:
Step1: Factor numerator and denominator
Numerator: (25x^{2}-64=(5x)^{2}-8^{2}=(5x - 8)(5x+8)) (using (a^{2}-b^{2}=(a - b)(a + b))). Denominator: (15x+24 = 3(5x+8)) So (f(x)=\frac{(5x - 8)(5x+8)}{3(5x+8)})
Step2: Simplify the function
Cancel out the common factor ((5x + 8)) (for (x\neq-\frac{8}{5})), we get (f(x)=\frac{5x - 8}{3}), (x\neq-\frac{8}{5})
Step3: Find the vertical asymptote
A vertical asymptote occurs where the denominator of the original (unsimplified) rational function is zero (after canceling non - zero common factors). Set the denominator of the original function (15x+24 = 0) Solve for (x): (15x=-24), so (x=-\frac{24}{15}=-\frac{8}{5})
Answer:
The vertical asymptote is (x =-\frac{8}{5})